Dynamical Modules in Metabolism, Cell and Developmental Biology Johannes Jaeger1,2 Nick Monk3*

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Dynamical Modules in Metabolism, Cell and Developmental Biology Johannes Jaeger1,2 Nick Monk3* Dynamical Modules in Metabolism, Cell and Developmental Biology Johannes Jaeger1,2 Nick Monk3* (1) Complexity Science Hub (CSH) Vienna, Josefstädter Straße 39, 1080 Vienna, Austria (2) School of Mathematics and Statistics, University of Sheffield, Hicks Building, Sheffield S3 7RH, UK * Corresponding author: [email protected], +44 114 222 3857 Abstract Modularity is an essential feature of any adaptive complex system. Phenotypic traits are modules in the sense that they have a distinguishable structure or function, which can vary (quasi-)independently from its context. Since all phenotypic traits are the product of some underlying regulatory dynamics, the generative processes that constitute the genotype- phenotype map must also be functionally modular. Traditionally, modular processes have been identified as structural modules in regulatory networks. However, structure only constrains, but does not determine the dynamics of a process. Here, we propose an alternative approach that decomposes the behaviour of a complex regulatory system into elementary activity-functions. Modular activities can occur in networks that show no structural modularity, making dynamical modularity more widely applicable than structural decomposition. Furthermore, the behaviour of a regulatory system closely mirrors its functional contribution to the outcome of a process, which makes dynamical modularity particularly suited for functional decomposition. We illustrate our approach with numerous examples from the study of metabolism, cellular processes, as well as development and pattern formation. We argue that dynamical modules provide a shared conceptual foundation for developmental and evolutionary biology, and serve as the foundation for a new account of process homology. Keywords modularity, dynamical systems, activity-function, regulatory networks, metabolic control, systems biology 1 1. Introduction: the Modular Epigenotype Organisms are characterised by distinguishable metabolic, physiological, developmental, morphological, and behavioral traits. These phenotypic traits are modular, able to evolve in a quasi-independent manner (Fig. 1A) (Lewontin, 1970, 1974). Herbert Simon (1962, 1973), was among the first to point out that such “near-decomposability” must be a fundamental and universal property of adaptive complex systems. Similar ideas are implicit in Stuart Kauffman’s notion of biological systems being poised on “the edge of chaos”—with modular “islands of chaos” among a large percolating component of “order” (Kauffman, 1993; see Jaeger & Monk, 2021, for a detailed discussion). By limiting the off-target (pleiotropic) effects of mutations, a modular architecture enables evolution through targeted selection for specific trait functions. Wings are adapted for flying, and legs for walking, because fore- and hind-limbs in aerial vertebrates can evolve quasi-independently: even though they share a common underlying structure, their shape and size can be altered in different directions (Fig. 1A). It is widely accepted today that trait modularity is a fundamental prerequisite for evolvability, defined as the capacity of a system to generate adaptive change (Dawkins 1989; Wagner & Altenberg 1996; Pigliucci 2008). Consistent with this, modular traits are ubiquitous in evolving organisms. They have been described at many different levels of organisation, from behavioral patterns to morphological characters to intercellular signaling pathways to cis-regulatory sequences involved in gene regulation (Schlosser & Wagner, 2004; Callebaut & Rasskin-Gutman, 2005). All biological traits are generated by some underlying regulatory dynamics (Jaeger et al., 2012; Jaeger & Monk, 2014; DiFrisco & Jaeger, 2021). These generative processes constitute the epigenotype of the organism (Waddington 1942, 1953, 1957; Goodwin, 1982; Oster & Alberch, 1982), which is often represented as a complex mapping from genotype to phenotype (Fig. 1B) (Burns 1970; Alberch 1991; Pigliucci 2010). If we are to properly understand the evolutionary sources of phenotypic variability—the substrate on which natural selection can act—we must study the structure of this mapping in a mechanistic manner (Jaeger et al., 2012; Jaeger & Monk, 2014, 2021; DiFrisco & Jaeger, 2019, 2020). What we are ultimately interested in are the causal processes that generate the variational properties of an adaptive and evolvable complex system. 2 Figure 1: Modularity of the genotype-phenotype map. (A) Phenotypic traits are modular. They are distinguishable in terms of either structure or function, and are able to evolve in quasi-independent ways. An example is the specialisation of vertebrate fore-limbs into arms and wings. (B) This implies that the processes that generate these traits—the epigenotype of the organism, which mediates the mapping from genotype to phenotype—must also be modular and dissociable (at least to some extent). Adapted from Jaeger & Monk (2021). See text for details. An important first step in this direction is to recognise that the quasi-independence of modular traits necessitates a modular structure of the underlying epigenotype (Fig. 1). The processes that generate traits must themselves be modular or dissociable (at least to some extent) for those traits to be able to vary quasi-independently (Needham, 1933). Therefore, it should be possible to map the source of variation in specific traits back to variability in specific modular subsystems of the epigenotype. This is not a trivial task and the precise nature of epigenetic modularity remains elusive, which is the problem we tackle in this paper. Elsewhere, we have developed a detailed theoretical account of dynamic modularity, which allows us to decompose the genotype-phenotype map into dissociable causal processes (Jaeger & Monk, 2021). Here, we briefly reiterate the main features of our account, illustrating it with a range of examples. In the next section, we start with an overview of the different types of modules one can identify in biological systems, comparing the advantages and drawbacks of 3 each kind of modular decomposition. We then continue by introducing our own dynamical and activity-based notion of modularity. The examples we provide represent regulatory processes from the fields of metabolism, cell and developmental biology. We conclude by discussing some of the wider implications of dynamical modularity, especially in evolutionary biology. 2. Kinds of Modules Modules, defined in the broadest sense, are parts, components, or subsystems of a larger system that possess a specific functional or structural identity (Moss, 2001). They must persist long enough to exert their effect (Callebaut, 2005). Modules themselves can consist of different kinds of entities: physical objects or structures, regulatory pathways or processes, and even types of behaviour as we shall show here. Modularity in complex systems is always a matter of degree (Wagner & Altenberg, 1996). On the one hand, all modules interact with their context— usually in a structured and hierarchical way. On the other hand, a system is modular to the extent to which its components operate according to their own intrinsically determined principles (Callebaut, 2005). In other words, modules exhibit internal causal cohesion coupled with a certain degree of autonomy from their context (Collier 1988, 2004; DiFrisco, 2018). This means that modules can be reused, since they are able to operate robustly across a certain range of circumstances (Mireles and Conrad, 2018). There are many different kinds of modules, and various methods to distinguish them. One approach classifies modules according to their biological context, e.g. developmental vs. evolutionary modules (Raff, 1996; Brandon, 1999; von Dassow & Munro, 1999; Bolker, 2000; Callebaut, 2005; Wagner et al. 2007). Another type of classification is based on what kind of entities a module is composed of: morphological modules consist of physical structures, developmental modules of processes (Winther, 2001; Callebaut, 2005). Here, we will focus on how modularity is relative to the specific way in which a system is decomposed into parts or subsystems. Each such decomposition provides a different perspective on a complex system (Wimsatt, 2007). Decompositions based on functional, regulatory, structural, or variational criteria result in modules with different properties and biological applications (Fig. 2) (see Jaeger & Monk, 2021, for a more detailed discussion). We will briefly review each of these in turn, before we move on to modules that are defined in terms of their dynamical behaviour. 4 Figure 2: Different kinds of modules. (A) Variational modules are distinguished by their ability to vary quasi-independently of the rest of the system. In the example shown here, the shape, size, and colour of ears and eyes can vary independently. These two morphological traits are therefore variational modules. (B) Functional modules are defined by the use-function their components contribute to. Shown here are the hierarchical layers of the Drosophila segmentation gene hierarchy. Genes in each layer cause similar kinds of phenotypes when mutated. They also show similarities in gene expression patterns (as shown here schematically). Downward arrows indicate hierarchical regulatory interactions; circular arrows indicate complex cross-regulation within each layer. (C) Structural motifs come in different flavours. Network motifs (left): small subnetworks that are statistically enriched above a given threshold (dashed line)
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